Off-shell, the gravitational theory has \((D^2-2D+2)\)  degrees of
freedom: \(D(D-1)/2\)  from the metric, \((D-1)(D-2)/2\)  from the 2-form and 1
from the scalar field. In contrast, each one of the gauge theories
has \((D-1)\)  degrees of freedom, which come from the gauge field. Taking the convolutive product of gauge theory fields then gives \((D-1)^2\)  degrees of freedom, which is one short of the gravitational theory. This problem is resolved in [1]} by including ghost fields to take care of the gauge symmetry, leading to BRST covariant theories. We discussed the BRST covariant formulation
of the gravitational theory in section , and we shall
review the BRST formulation of Yang-Mills theory in REF .
The gravitational theory, then, has fields
\((h_{\mu \nu }, B_{\mu \nu }, \varphi , c_\mu , \bar{c}_\mu , d_{\mu },\bar{d}_\mu , d, \bar{d}, \eta )\)  and the two Yang-Mills theories
have fields \((A_{\mu }, c, \bar{c})\)  and \(({{\widetilde{A}}}_{\mu }, {{\tilde{c}}}, \bar{{{\tilde{c}}}})\) 
(see section REF ). The convolutive product of the ghost fields of the Yang-Mills theories gives 4 degrees of freedom, which accounts for the degrees of freedom in \((d, \bar{d}, \eta )\)  and a combination of \(h\)  and \(\varphi \) , which we can simply think of as accounting for \(\varphi \) . The convolutive product of the ghost sector and the gauge fields then contributes \(4(D-1)\)  degrees of freedom, which account for the gravitational ghost fields \((c_\mu , \bar{c}_\mu , d_{\mu }, \bar{d}_\mu )\) . Finally, now, the \((D-1)^2\)  degrees of freedom coming from the convolutive product of the gauge fields accounts for the degrees of freedom in \(h_{\mu \nu }\)  and \(B_{\mu \nu }.\)  In addition, the introduction of the ghost fields allows one to disentangle the degrees of the freedom in \(h\)  and \(\varphi \) .
